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STAT1203 PROJECT 2:
WASHI DUE: WEEK 13 MON 5PM Like many industrialised nations, Japan has been experiencing the issue of population aging to an unprecedented degree. The country’s population fell by a record of over 480,000 in 2020 and the pace of this decline is predicted to accelerate each year until the 2060s. The hardest hit in the population decline is in the rural areas, where young people left for cities like Tokyo and Osaka to look for jobs. As a result, federal and local governments have numerous iju¯1 programs, with monetary incentives up to millions of yen, to attract young people to move to the rural areas. The Adachis are a young couple drawn by these incentives and the opportunity to live a less stressful life. A few years ago, they moved back to a village near where Glenda and John made their deliveries. They run a farm growing organic produce. One of the crops they grow is ko¯zo2, a main ingredient for making washi paper. (1) The long fibres of kozo create incredibly strong and durable papers. This year, they are trying a new kozo variety and they want to know how it compares to the variety they planted in previous years. A sample of washi papers is made using the new variety. The strength of the papers is tested by applying a particular force to each piece and determining whether a tear (T ) is observed: T T¯ T¯ T¯ T¯ T¯ T¯ T¯ T¯ T¯ T Current Test outcomes Let X be the number of tests (including the current one) between two paper tears (So X = 10 here). They approached Glenda and John for an answer. (a) Glenda and John identified a known probability model (distribution) for the data. What is that distribution? What are the assumptions? What is(are) the meaning(s) of the parameter(s) in the model? (b) State E(X) and var(X) for the model in (a). 1relocation 2a mulberry-like shrub 2021-22 Term 1 2 (c) Based on the Adachis’ experience on the variety they planted in previous years, 1 in 6 pieces of paper under test would result in a tear. Combining this information and the observed data, Glenda and John carried out a 5% significance test. Repeat their workings. (d) Glenda and John were a little surprised at the result, so they ask the Adachis to carry out a few more tests. With more tests, including the first observation, the data look like: (x1, x2, ..., x32) 1, 8, 1, 5, 12, 39, 6, 4, 2, 13, 14, 5, 5, 10, 10, 2, 3, 2, 1, 17 10, 3, 11, 3, 4, 17, 8, 10, 23, 1, 6, 4 Glenda and John repeated the analysis by assuming the sample size n is “large”. It is known that the MLE of the parameter in the model is 1 x¯ where x¯ = 1 n ∑n i=1 xi. So they sent Aloysius a message for help. He comes back with an answer. Repeat the works of Glenda, John and Aloysius. Hint: Aloysius assumes, using the old variety papers, in every 6 tests, one would result in a tear. (2) The kozo the Adachis planted are used by the village’s artisans to produce ultra- thin paper for calligraphy and suiboku-ga, or ink wash painting. The paper is extremely delicate and fragile. The main idea behind suiboku-ga is creating a gradient from light to dark. Objects in the background should be a light gray while subjects in the forefront appear darker. Production of the paper involves a lot of trial and error, so that the paper will soak up the ink in the right way. Over time the artisans discovered that the frequency of paper with flaws, S, in a stack of 200 sheets of papers, follow the distribution: j 0 1 2 3 4 5 6 7 8 9 10 P(S = j) 0.098 0.124 0.138 0.208 0.168 0.112 0.056 0.03 0.031 0.027 0.008 (a) Describe and summarise the distribution of S using methods we learned. (b) The artisans wish to investigate whether a new method would reduce the average number of flaws. To do that, they produce some papers using the new method, and record the following number of flaws in 25 stacks of 200 papers: 4, 2, 2, 3, 2, 3, 4, 2, 0, 3, 0, 3, 2, 3, 6, 0, 0, 1, 3, 3, 0, 1, 0, 2, 4. By making reasonable assumptions, carry out a 5% significance test to determine whether the new method reduces the average number of flaws compared to the old method. 2021-22 Term 1 3 (3) One of the initiatives of the papermakers is to bring old paper back to life. The old paper, koshi is boiled and mixed with kozo, and then beaten to produce the material for making new paper. The quality of the end product, Y (on a scale from 0-10), depends on a number of conditions.